Porous materials are widely applied in science and engineering with applications including filtration devices, acoustic and thermal insulation materials, low-k dielectrics in microelectronic devices, ultra-light materials, catalysts, ion exchange materials, optical coatings, photovoltaics, sensing devices and many more. Porous materials consist of a skeletal portion often referred to as matrix or host material and pores which are filled with a fluid, either liquids or gases. The medium is characterized by its total porosity, i.e., the volume fraction of empty or filled space in relation to the overall material volume. Depending on the applications, additional characteristics can be of interest such as the pore diameter and the specific surface area, i.e., the total surface area including pores relative to a unit of mass or volume. The overall mechanical (eg. tensile strength, strain), optical (high-frequency permittivity), and magneto-electric (low-frequency permittivity, conductivity, permeability etc.) properties of the porous material are a result of the combined properties of its constituents and can often be approximated using effective medium theories. Due to the complicated microstructure, these effective properties often vary within the material resulting, for example, in anisotropic optical and electronic properties, or gradients in pore size and overall porosity.
Porous materials are classified according to IUPAC notation as microporous for average pore diameters below 2 nm, mesoporous for pore diameters between 2-50 nm, and macroporous for pore diameters above 50 nm. The most common technique to characterize the overall porosity and pore size distribution in mesoporous and microporous samples is N2 adsorption porosimetry which determines adsorption and desorption isotherms for bulk porous samples of large volume by directly measuring the weight increase/decrease due to adsorption/desorption of liquid (N2) on/from the pore walls (for macroporous samples, high-pressure mercury porosimetry is commonly applied, but macroporous samples are not topic of this investigation). The calculation of the pore size distribution in mesoporous samples assumes progressive (instantaneous) filling and emptying of porous systems of a specific pore size at adsorbate (here N2) partial pressures below the saturation vapor pressure of that solvent for flat surfaces, P0. The dependence of the relative pressure P/P0 at which condensation in pores occurs on the meniscus curvature is given by the Kelvin equation:
                              ln          ⁡                      (                          P                              P                0                                      )                          =                  -                                    f              ⁢                                                          ⁢              γ              ⁢                                                          ⁢                              V                m                            ⁢              cos              ⁢                                                          ⁢              θ                                                      r                k                            ⁢              RT                                                          (        1        )            where γ and Vm are the surface tension and molar volume of the liquid solvent (adsorbate), θ is the contact angle of the liquid solvent on a non-porous surface of the host material, R is the molar gas constant, T is the temperature in K, and f is a shape factor equal to 1 for slit-shaped pores and equal to 2 for cylindrical pores, respectively. The quantity rK is the Kelvin radius, which is related to the actual pore radius, rp, by rp=rk+t where t is the thickness of the layer adsorbed on the pore walls shortly before condensation occurs. The parameter t describes the observation that some amount of solvent will adsorb to any surface and form thin (partial) layers in dependence of the relative pressure and can be either determined by measuring the thickness of the adsorbed solvent vs. relative pressure on a flat non-porous surface of the same material or estimated from the Brunauer, Emmet, Teller (BET) equation. FIG. 1 shows the typical hysteresis loop of the determined condensed solvent volume within the pores of a mesoporous SiO2 in dependence of the relative solvent pressure which is a result of different effective radius of curvature of the condensed liquid meniscus during adsorption and desorption.
Microporous samples with pore diameters smaller than 2 nm show significant variation of the optical properties of the sample for relative pressures P/P0<0.1. The theory describing the variation of condensed solvent volume within the micropores in dependence of the relative solvent pressure is based on the assumption of micropore volume filling rather than layer-by-layer adsorption on pore walls (Dubinin-Radushkevitch theory) which will not be further outlined here. The pore size distribution for a porous sample can be determined by analyzing the dependence of the condensed solvent volume on the relative pressure P/P0. Under the assumption that all pores of the same radius are filled or emptied more or less instantaneous when the condition of the Kelvin equation are met, then the pore size distribution can be determined by calculating the derivate of the condensed solvent volume in the pores vs. relative pressure. The corresponding pore radii are calculated from the relative pressure values P/P0 by applying Eq. (1). (Note at this point, that FIG. 1 shows Typical hysteresis of the adsorbed solvent (water) volume for an adsorption/desorption cycle on a mesoporous SiO2 film on Si, and FIG. 2 shows Pore size distribution for the example of porous SiO2 on Si shown in FIG. 1).
Pore Size Analysis from Spectroscopic Ellipsometry or Intensity Data Using the Established Lorentz-Lorenz Equation Approach
The basic idea of using spectroscopic ellipsometry or intensity data for determining structural properties of porous samples is to monitor the change of optical properties due to condensation of solvent within the pores rather than measuring absolute weight of condensed solvent which would be difficult for thin film samples. Spectroscopic ellipsometry accurately determines the optical properties of a sample with change of the relative solvent pressure and can simultaneously monitor variations of the thickness for thin film samples. Spectroscopic ellipsometry is an indirect technique which requires an optical model for which relevant parameters are varied during a regression analysis in order to best match the experimental data. The variation of the optical properties of a porous sample can be related to the amount of condensed solvent within the pores by applying effective medium theories. Several different models were derived based on certain assumptions on microstructural mixing of constituents. For pore diameters much smaller than the wavelength of the probing light beam, good agreement between the derived fractions of skeletal material, void, and potential other constituents (such as a partial fill by a liquid) with the same quantities determined by alternative experimental techniques has been demonstrated. To determine the pore size distribution, experimental spectroscopic ellipsometry or intensity data has to be acquired over a wide relative pressure range in order to obtain the characteristic hysteresis loop observed in a standard adsorption measurement. An appropriate procedure to obtain the desired condensed solvent volume within the pores from the spectropic ellipsometry, (or intensity), data for different relative solvent pressure values has to be chosen.
A procedure based on the Lorentz-Lorenz effective medium theory is well documented in literature. The Lorentz-Lorenz model is an extension of the Clausius-Mosotti equation for the polarizability of spherical particles in air assuming multiple particle species which are homogeneously mixed on the microscopic scale:
                                          ɛ            -            1                                ɛ            +            2                          =                                            f              a                        ⁢                                                            ɛ                  a                                -                1                                                              ɛ                  a                                +                2                                              +                                    (                              1                -                                  f                  a                                            )                        ⁢                                                            ɛ                  b                                -                1                                                              ɛ                  b                                +                2                                                                        (        2        )            
For effective mediums consisting of a skeletal material with some void space neither the assumption of a microscopic mix nor the matrix being air is a good assumption. Therefore, this model is typically not used to analyze ellipsometric or intensity data. However, the equation was adopted in the porosimetry field since it can be used to derive a very simple equation to describe the condensed solvent volume in porous samples at a specific relative pressure. The polarizability P for the three cases of a solid film (ns), empty porous films (ne), (see FIG. 9A), filled with air (n0=1), (see FIG. 9B), and porous film (nrel) partially filled with solvent (nsol), (see FIG. 9C), are given by, respectively:
            P      =                                    n            s            2                    -          1                                      n            s            2                    +          2                      ;              P      =                                                  n              e              2                        -            1                                              n              e              2                        +            2                          =                                                            V                1                            ⁢                                                                    n                    0                    2                                    -                  1                                                                      n                    0                    2                                    +                  2                                                      +                                          (                                  1                  -                                      V                    1                                                  )                            ⁢                                                                    n                    s                    2                                    -                  1                                                                      n                    s                    2                                    +                  2                                                              =                                    (                              1                -                V                            )                        ⁢                                                            n                  s                  2                                -                1                                                              n                  s                  2                                +                2                                                          ;              P      ⁡              (                  p                      p            0                          )              =                                        n            rel            2                    -          1                                      n            rel            2                    +          2                    =                                    V            1                    ⁢                                                    n                0                2                            -              1                                                      n                0                2                            +              2                                      +                              V            2                    ⁢                                                    n                sol                2                            -              1                                                      n                sol                2                            +              2                                      +                              (                          1              -                              V                1                            -                              V                2                                      )                    ⁢                                                                      n                  s                  2                                -                1                                                              n                  s                  2                                +                2                                      .                                              (Where V . . . =Total Porosity).        
Under the assumption that the sum of the empty volume V1 and the condensed solvent volume V2 in the partially filled pores equals the total porosity V of the empty pores, the last two equations can be combined and rearranged in order to determine the solvent volume V2 in the pores at arbitrary filling state, i.e., at any relative pressure (P/P0):
                                          V            2                    ⁡                      (                          P                              P                0                                      )                          =                              (                                                                                n                    rel                    2                                    -                  1                                                                      n                    rel                    2                                    +                  2                                            -                                                                    n                    e                    2                                    -                  1                                                                      n                    e                    2                                    +                  2                                                      )                    /                                                    n                sol                2                            -              1                                                      n                sol                2                            +              2                                                          (        3        )            This equation only depends on the homogenous layer refractive index determined for the case of empty pores (ne at P/P0=0), the known solvent refractive index nsol, and the homogenous layer refractive index nrel determined at that specific relative pressure. nrel and ne refer to the refractive indices derived from a model analysis under the assumption that the porous material can be described as one homogeneous solid layer of equivalent optical properties.
The ellipsometric or intensity data analysis is performed step-by-step for each time slice of an adsorption or desorption cycle (scan over wide relative pressure range. An appropriate optical model has to be chosen to describe the optical properties of the porous layer (In many cases, a simple Cauchy model for transparent films will be sufficient to match the data). The user has to identify a refractive index value to use for the condensed solvent volume calculation according to Eq. (3). In most cases, this will be the refractive index value at a specific wavelength, i.e., the spectroscopic nature of the experiment is completely ignored. For this approach, the sample has to be isotropic. Further, all time slices are modeled independent of any other time slice. Therefore, noise or non-idealities not accounted for in the ellipsometric model will directly transfer to noise in the determined condensed solvent volume. The total porosity is adopted from the resulting solvent volume curve vs. relative pressure by extrapolation to the relative pressure P/P0=1 under the assumption that all pores are filled at that pressure. Potential isolated pore volumes (not accessible by solvent) are ignored. The pore size distribution is determined from the solvent volume vs. relative pressure plots.
Advantages of the Lorentz-Lorenz equation formalism:                Skeletal material refractive index does not need to be known;        Very simple, only needs single-wavelength refractive index values;Disadvantages:        Based on invalid assumptions on microscopic nature of the film;        Randomly uses single-wavelength refractive indices despite analyzing spectroscopic data (which one to choose?);        Only applicable to isotropic and homogenous materials, however, most porous films show out-of-plane anisotropy (pore shape) or graded properties (porosity)        Assumes completely empty pores at P/P0=0 and filled pores at P/P0=1;        No access to isolated pore volume and refractive index of the skeletal material.        
Despite being a fairly simple approach, useful data can be obtained by using the Lorentz-Lorenz equation formalism. However, in many cases only mediocre match between model and experimental data will be achieved. In this case, it is not possible to predict what error is present in the determined solvent volume considering the fact that the model is not capable of matching the experimental data.
Known patents in the area are U.S. Pat. No. 9,423,447 to Kiermasz, and U.S. Pat. No. 5,248,614 to Wang.
Even in view of the prior art, need remains for a method of determining pore size and distribution in an effective thin layer, or a surface region of a ocmi-infinite infinite bulk substrate based on applying a Bruggeman effective medium model that follows physically motivated and accurate ellipsometric, (or intensity) data analysis. In particular need remains for a method which can account for the anisotropic nature of an effective layer on a substrate, can access pore volume, account for graded sample properties and does not rely on assumptions about the pore filling state at any given relative solvent pressure.